### Approximate Solution of the Kersten-Krasil’shchik Coupled Kdv-Mkdv System via Reduced Differential Transform Method

Authors: Ahmed F. Qasim1 & Mohammed O. Al-Amr2
1&2Department of Mathematics, College of Computer Sciences and Mathematics, University of Mosul, Iraq

Abstract:  In this paper, the approximate solution of the Kersten-Krasil’shchik coupled KdV-mKdV system is obtained by using the reduced differential transform method (RDTM). This system is regarded as a classical super-extension of the KdV equation. The obtained results are compared with the exact solutions to show the efficiency and reliability of the proposed method which can be extended to solve a large variety of nonlinear partial differential equations.

Keywords: Reduced Differential Transform Method, Kersten-Krasil’shchik Coupled KdV-mKdV System, Partial Differential Equations

References

Al-Amr, M. O. & El-Ganaini, S. (2017). New exact traveling wave solutions of the (4+1)-dimensional Fokas equation. Comput. Math. Appl., 74, 1274-1287. http://dx.doi.org/10.1016/ j.camwa.2017.06.020

Al-Amr, M. O. (2014). New applications of reduced differential transform Method. Alexandria Eng. J., 53, 243–247. http://dx.doi.org/10.1016/j.aej.2014.01.003

Al-Amr, M.O. (2015). Exact solutions of the generalized (2+1)-dimensional nonlinear evolution equations via the modified simple equation method. Comput. Math. Appl., 69, 390–397. http://dx.doi.org/10.1016/j.camwa.2014.12.011

Al-Rozbayani, A. M., & Al-Amr, M. O. (2013). Discrete Adomian Decomposition Method     for Solving Burgerʼs−Huxley Equation. Int. J. Contemp. Math. Sciences, 8, 623-631. http://dx.doi.org/10.12988/ijcms.2013.3570

Al-Sawoor, A. J., & Al-Amr, M. O. (2012). Numerical Solution of a Reaction-Diffusion System with Fast Reversible Reaction by Using Adomian’s Decomposition Method and He’s Variational Iteration Method. Al-Rafidain J. Comput. Sci. Math., 9, 243-257.

Al-Sawoor, A. J., & Al-Amr, M. O. (2013). Reduced differential transform method for the generalized Ito system, Int. J. Enhanc. Res. Sci. Technol. Eng., 2, 135-145.

Al-Sawoor, A. J., & Al-Amr, M. O. (2014). A new modification of variational iteration method for solving reaction-diffusion system with fast reversible reaction. J. Egyptian Math. Soc., 22, 396–401. http://dx.doi.org/10.1016/j.joems.2013.12.011

Gepreel, K. A., Omran S., & Elagan S. K. (2011). The traveling wave solutions for some nonlinear PDES in mathematical physics. Applied Mathematics, 2, 343-347. http://dx.doi.org/10.4236/ am.2011.23040

He, J. H. (2003). Homotopy perturbation method: a new nonlinear analytical technique. Appl. Math. Comput., 135, 73–79. https://dx.doi.org/10.1016/S0096-3003(01)00312-5

Hon, Y. C., & Fan, E. G. (2004). Solitary wave and doubly periodic wave solutions for the Kersten-Krasil’shchik coupled KdV-mKdV system. Chaos, Solitons and Fractals, 19, 1141–1146. http://dx.doi.org/10.1016/S0960-0779(03)00302-3

Kalkanli, A. K., Sakovich, S. Y., & Yurdusen, I. (2003). Integrability of Kersten-Krasil’shchik coupled KdV-mKdV equations: singularity analysis and Lax pair. J. Math. Phys., 44, 1703-1708. http://dx.doi.org/10.1063/1.1558903

Kersten, P., & Krasil’shchik, J. (2000). Complete integrability of the coupled KdV-mKdV system. Adv. Stud. Pure Math., 89, 151-171.

Keskin, Y., & Oturanc, G. (2009). Reduced differential transform method for partial differential equations. Int. J. Nonlinear Sci. Numer. Simul., 10, 741–749. https://dx.doi.org/10.1515/ IJNSNS.2009.10.6.741

Keskin, Y., & Oturanc, G. (2010). Reduced differential transform method for generalized KdV equations. Math. Comput. Appl., 15 (3), 382–393. https://doi.org/10.3390/mca15030382

Qin, Y., Gao, Y.T., Yu, X., & Meng, G.Q. (2012). Bell polynomial approach and N-soliton solutions for a coupled KdV-mKdV system. Commun. Theor. Phys., 58, 73-78. https://dx.doi.org/10.1088/0253-6102/58/1/15

Rui, W., & Qi, X. (2016). Bilinear approach to quasi-periodic wave solutions of the Kersten-Krasil’shchik coupled KdV-mKdV system. Boundary Value Problems, 2016:130. http://dx.doi.org/10.1186/s13661-016-0634-3.

Saeed, R. K., & Muhammad, R. S. (2016). Solving Coupled Hirota System by Using Reduced
Differential Transform Method. International Journal of Mathematical Engineering and Science, 5, 16-30.

Saeed, R. K., & Mustafa, A. A. (2017). Numerical solution of Fisher–KPP equation by using reduced differential transform method. AIP Conf. Proc., 1888, 020045-1–020045-11. http://dx.doi.org/10.1063/1.5004322.

Wazwaz, A. M. (2004). A sine–cosine method for handling nonlinear wave equations. Math. Comput. Modelling, 40, 499–508. https://dx.doi.org/10.1016/j.mcm.2003.12.010

Wazwaz, A. M. (2009). Partial differential equations and solitary waves theory. Beijing: Higher Education Press.

Whitham, G. B. (1974). Linear and nonlinear waves. New York, NY: Wiley.